The AMA doesn't directly govern the number of medical schools in this country. That would be the LCME (Liaison Committee on Medical Education), which is sponsored by the AMA and American Association for Medical Colleges.
It's also important to note that the number of medical schools isn't really artificially limited by the LCME, as they will accredit most any medical school that meets the accreditation standards.
On top of that, medical students in the US have to undergo a residency before becoming fully licensed physicians. These residencies are accredited by the Accreditation Council for Graduate Medical Education (ACGME), a separate organization.
However, while the ACGME can accredit these programs, the majority of funding for training resident physicians comes from the federal government (department of health and human services, center for medicare and medicaid services), and is capped by the Balanced Budget Act of 1997 (modified in 1999 to accommodate the rural physician shortage). Today, this costs the federal government around $11 billion each year for all of the residents in the country.
The real question, then, is who we can point to as the bottleneck for new doctors in this country. I found this article enlightening, but it may be behind a paywall:
https://www.ncbi.nlm.nih.gov/pubmed/28919221